I can lead this off with Judaism. The thing to keep in mind is that what we now call the Jewish religion was originally what we could call "Yahwehism". There are two major distinctions between Yahwehism and Judaism that are key.

First, the Yahweh cult acknowledged the existence of foreign deities, whereas Judaism does not. The Hebrew tribes existed in a world that thought along the lines of "You have you god(s), we have our gods(s). Our gods rule our land, your gods rule yours." If two rival states fought against each other, it was said that their respective gods were also engaged in the fight as well, thus gods could be victorious or defeated. Judaism denies the existence of all other gods, and God is the single god of the universe. Yahweh also resided in his temple at Jerusalem, which was commissioned by David and built by Solomon, hence why we call it "Solomon's Temple".

Second, geography and politics. According to tradition, the Kingdom of Israel was founded around 1000BCE as a single, united kingdom. Not long after, it split in two with Israel in the north and Judah in the south. Israel was conquered in 722BCE by the Assyrians and Judah fell in 586BCE. Judah just also so happened to be where Jerusalem and the Temple of Solomon was. The temple and the conquest by Babylon are key here because of two things:

1. The temple, the center of the Yahweh cult, was completely destroyed.

2. The people of Judah were exiled, primarily to Babylon. Biblical/Jewish historians often call this the "Babylonian Exile" or "Babylonian Capitivity" (By the rivers of Babylon where we sat down...)

So basically, not only was the Kingdom of Israel destroyed, but their god, Yahweh, was "defeated". But wait, part of the Hebrew dogma was that they were God's chosen, protected people. Yahweh had made a "unbreakable covenant" with his people to protect them. The Hebrews/Israelites in exile had a major existential question to confront: "If our god is all-powerful, why have we been defeated and taken away from the promised land?"

In short, the answer to this was that God was using the Assyrians/Babylonians as his tools, and the exile of the Hebrews was, in fact, part of God's plan all along! God IS in fact all-powerful and the god of the universe, and just because the Hebrews aren't in the promised land, that isn't changed. In God's covenants with previous church fathers (Abraham, David), God specifically mentions that should their descendants not hold up their end of the bargain, God would punish them. Thus, Assyria and Babylon were God's punishments for Israelite unfaithfulness to him.

But then there's a new problem: where (and how) does God get worshipped? Enter the idea of the Synagogue - wherever a Hebrew/now more accurately called Jewish community is, God is. God is present everywhere, thus his temple is no longer necessary for the now-Jews to have their identity and religion. Sure, the temple would be rebuilt/destroyed/rebuilt/destroyed finally by Rome, it did not have the same religious weight as it did during the time of the Kingdom.

What essentially happens, and there is a very strong parallel in Christianity, is that a local cult changed in a way that maintained its unified identity while dissociating itself (for the most part) with the geographical area with which it was associated. The LOCALIZED Hebrew religion of Yahweh became the UNIVERSAL religion of Judaism. Universal religions by their nature spread much more easily precisely because there can be a community of that religion anywhere. This same exact pattern would play out in Europe as pagans converted to Christianity. Removed for speculating.

This tension is played out extensively in the Hebrew Bible, specifically in 2 Samuel, Kings, and the earlier prophetical books.

Sources:

The New Oxford Annotated Study Bible, introductions to 2 Samuel, 1/2 Kings

The Bible and the Ancient Near East, by Cyrus Gordon and Gary Rendsburg. Published by W.W. Norton & Company.

You might be interested in the perspective of sociologist Michael Mann, who looks at the rise of "religions of the book" in Volume 1 of The Sources of Social Power. Mann groups Islam and Christianity with Buddhism and Hinduism as religions that are all concerned with salvation, which he defines as "the goal of relief from earthly sufferings through some kind of systematic moral life plan available to all, regardless of class or particularistic identity." (p. 301-302)

A large part of the appeal of these faiths is that they all fostered a sense of far-reaching community within an imperial context. An adherence to a religion created a social bond between individuals that transcended ties of blood and social class, which made it possible for large numbers to live together in peace.

On the other hand, Christianity and Islam also demanded a degree of exclusivity in their adherents' beliefs:

Christianity was a religion of the book. The book contained dogma. By accepting the dogma one became a Christian. All can join; it is an act of free will. But what if one's sense of truth is different, and one prefers, say, a more elaborate Greek philosophy, or the republican virtues of the pagan gods, or the ecstasy of the mystic cult? Christianity, like Zoroastrianism and Islam, defined the essence of humanity as rational acceptance of its truth. Therefore rejection of the faith renders one nonhuman. This characteristic of religions of the book detracts from their universalism. Earlier religions had tended to either exclude the masses from participation in higher truth or accept that different ascriptive groups had their own truths. If another group was thought to lack humanity, this had nonreligious sources. Now religion defined and restricted humanity. (p. 330)

So both the Christian ecumene and the Muslim umma created a community open to all individuals, regardless of whether they had kinship ties, but also required that they adhere to a set of ideological principles. Because both religions were built on literacy, these principles had a certain consistency (as opposed to oral religions, which tend to be unstable and eclectic). Combined with the promise of salvation, this gave both religions a great deal of appeal - particularly within the imperial context of the Roman empire, or in a social context in which groups were divided through tribal feuds.

As Mann puts it, all these salvationist religions "gave a considerable impetus to the transcendent exchange of messages, and therefore of controls, across hierarchical levels, genders, ethnic divisions, and state boundaries. The most common effects were on different classes and 'peoples' brought into a common sense of identity. This was also a profound change, for it led potentially to the mobilization of the masses. Hitherto... societies had been strongly federal. Power had been divided among various hierarchical and regional coordinating levels. The masses had not usually been within direct reach of the highest, most centralized levels of power. The beliefs of the masses had not been relevant to the exercise of macrosocial power. Now the masses and the centers of power were ideologically connectible." (p. 365)

Anyway, I'm not sure if I'm doing justice to the complexity of Mann's argument - and if you're curious about the details, see chapters 10 & 11 of his book - but I think it's this socially cross-cutting appeal of Abrahamic religions that allowed them to expand so quickly.